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Elementary Algebra 2e

9.2 Simplify Square Roots

Elementary Algebra 2e9.2 Simplify Square Roots
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:

  • Use the Product Property to simplify square roots
  • Use the Quotient Property to simplify square roots
Be Prepared 9.4

Before you get started take this readiness quiz.

Simplify: 8017680176.
If you missed this problem, review Example 1.65.

Be Prepared 9.5

Simplify: n9n3n9n3.
If you missed this problem, review Example 6.59.

Be Prepared 9.6

Simplify: q4q12q4q12.
If you missed this problem, review Example 6.60.

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that 5050 is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use Figure 9.2.

But what if we want to estimate 500500? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

Simplified Square Root

aa is considered simplified if aa has no perfect square factors.

So 3131 is simplified. But 3232 is not simplified, because 16 is a perfect square factor of 32.

Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that (ab)m=ambm(ab)m=ambm. The corresponding property of square roots says that ab=a·bab=a·b.

Product Property of Square Roots

If a, b are non-negative real numbers, then ab=a·bab=a·b.

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in Example 9.12.

Example 9.12

How To Use the Product Property to Simplify a Square Root

Simplify: 5050.

Try It 9.23

Simplify: 4848.

Try It 9.24

Simplify: 4545.

Notice in the previous example that the simplified form of 5050 is 5252, which is the product of an integer and a square root. We always write the integer in front of the square root.

How To

Simplify a square root using the product property.

  1. Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
  2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Step 3. Simplify the square root of the perfect square.

Example 9.13

Simplify: 500500.

Try It 9.25

Simplify: 288288.

Try It 9.26

Simplify: 432432.

We could use the simplified form 105105 to estimate 500500. We know 5 is between 2 and 3, and 500500 is 105105. So 500500 is between 20 and 30.

The next example is much like the previous examples, but with variables.

Example 9.14

Simplify: x3x3.

Try It 9.27

Simplify: b5b5.

Try It 9.28

Simplify: p9p9.

We follow the same procedure when there is a coefficient in the radical, too.

Example 9.15

Simplify: 25y5.25y5.

Try It 9.29

Simplify: 16x716x7.

Try It 9.30

Simplify: 49v949v9.

In the next example both the constant and the variable have perfect square factors.

Example 9.16

Simplify: 72n772n7.

Try It 9.31

Simplify: 32y532y5.

Try It 9.32

Simplify: 75a975a9.

Example 9.17

Simplify: 63u3v563u3v5.

Try It 9.33

Simplify: 98a7b598a7b5.

Try It 9.34

Simplify: 180m9n11180m9n11.

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify 25+14425+144 we must simplify each square root separately first, then add to get the sum of 17.

The expression 17+717+7 cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

Example 9.18

Simplify: 3+323+32.

Try It 9.35

Simplify: 5+755+75.

Try It 9.36

Simplify: 2+982+98.

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

Example 9.19

Simplify: 44824482.

Try It 9.37

Simplify: 1075510755.

Try It 9.38

Simplify: 64536453.

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Example 9.20

Simplify: 964964.

Try It 9.39

Simplify: 25162516.

Try It 9.40

Simplify: 49814981.

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Example 9.21

Simplify: 45804580.

Try It 9.41

Simplify: 75487548.

Try It 9.42

Simplify: 9816298162.

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, aman=amn,a0aman=amn,a0.

Example 9.22

Simplify: m6m4m6m4.

Try It 9.43

Simplify: a8a6a8a6.

Try It 9.44

Simplify: x14x10x14x10.

Example 9.23

Simplify: 48p73p348p73p3.

Try It 9.45

Simplify: 75x53x75x53x.

Try It 9.46

Simplify: 72z122z1072z122z10.

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

(ab)m=ambm,b0(ab)m=ambm,b0

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Quotient Property of Square Roots

If a, b are non-negative real numbers and b0b0, then

ab=abab=ab

Example 9.24

Simplify: 21642164.

Try It 9.47

Simplify: 19491949.

Try It 9.48

Simplify: 28812881.

Example 9.25

How to Use the Quotient Property to Simplify a Square Root

Simplify: 27m319627m3196.

Try It 9.49

Simplify: 24p34924p349.

Try It 9.50

Simplify: 48x510048x5100.

How To

Simplify a square root using the quotient property.

  1. Step 1. Simplify the fraction in the radicand, if possible.
  2. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
  3. Step 3. Simplify the radicals in the numerator and the denominator.

Example 9.26

Simplify: 45x5y445x5y4.

Try It 9.51

Simplify: 80m3n680m3n6.

Try It 9.52

Simplify: 54u7v854u7v8.

Be sure to simplify the fraction in the radicand first, if possible.

Example 9.27

Simplify: 81d925d481d925d4.

Try It 9.53

Simplify: 64x79x364x79x3.

Try It 9.54

Simplify: 16a9100a516a9100a5.

Example 9.28

Simplify: 18p5q732pq218p5q732pq2.

Try It 9.55

Simplify: 50x5y372x4y50x5y372x4y.

Try It 9.56

Simplify: 48m7n2125m5n948m7n2125m5n9.

Section 9.2 Exercises

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

53.

2727

54.

8080

55.

125125

56.

9696

57.

200200

58.

147147

59.

450450

60.

252252

61.

800800

62.

288288

63.

675675

64.

12501250

65.

x7x7

66.

y11y11

67.

p3p3

68.

q5q5

69.

m13m13

70.

n21n21

71.

r25r25

72.

s33s33

73.

49n1749n17

74.

25m925m9

75.

81r1581r15

76.

100s19100s19

77.

98m598m5

78.

32n1132n11

79.

125r13125r13

80.

80s1580s15

81.

200p13200p13

82.

128q3128q3

83.

242m23242m23

84.

175n13175n13

85.

147m7n11147m7n11

86.

48m7n548m7n5

87.

75r13s975r13s9

88.

96r3s396r3s3

89.

300p9q11300p9q11

90.

192q3r7192q3r7

91.

242m13n21242m13n21

92.

150m9n3150m9n3

93.

5+125+12

94.

8+968+96

95.

1+451+45

96.

3+1253+125

97.

1024210242

98.

88048804

99.

3+9033+903

100.

15+75515+755

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

101.

49644964

102.

1003610036

103.

1211612116

104.

144169144169

105.

72987298

106.

75127512

107.

4512545125

108.

300243300243

109.

x10x6x10x6

110.

p20p10p20p10

111.

y4y8y4y8

112.

q8q14q8q14

113.

200x72x3200x72x3

114.

98y112y598y112y5

115.

96p96p96p96p

116.

108q103q2108q103q2

117.

36353635

118.

1446514465

119.

20812081

120.

2119621196

121.

96x712196x7121

122.

108y449108y449

123.

300m564300m564

124.

125n7169125n7169

125.

98r510098r5100

126.

180s10144180s10144

127.

28q622528q6225

128.

150r3256150r3256

129.

75r9s875r9s8

130.

72x5y672x5y6

131.

28p7q228p7q2

132.

45r3s1045r3s10

133.

100x536x3100x536x3

134.

49r1216r649r1216r6

135.

121p581p2121p581p2

136.

25r864r25r864r

137.

32x5y318x3y32x5y318x3y

138.

75r6s848rs475r6s848rs4

139.

27p2q108p5q327p2q108p5q3

140.

50r5s2128r2s550r5s2128r2s5

Everyday Math

141.
  1. Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify 288288 to determine the length and the width of his garden. Round to the nearest tenth of a foot.
  2. Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify 28852885 to determine the length and width of the new garden. Round to the nearest tenth of a foot.
142.
  1. Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify 64166416 to determine the number of seconds it takes for the sunglasses to reach the ground.
  2. Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify 1441614416 to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

143.

Explain why x4=x2x4=x2. Then explain why x16=x8x16=x8.

144.

Explain why 7+97+9 is not equal to 7+97+9.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and three rows. The columns are labeled, “I can…,” “confidently,” “with some help,” and “no—I don’t get it!” The rows under “I can…” Read, “use the Product Property to simplify square roots.,” and “use the Quotient Property to simplify square roots.” The other rows unders the other columns are blank.

After reviewing this checklist, what will you do to become confident for all objectives?

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